Preheat oven to 450F/230C. Wash mushrooms and let drain (or spin dry in salad spinner, which is what I did.) While mushrooms are drying, trim ends of beans and cut beans in half so you have bite-sized pieces. (An easy way to trim them is to gather a small handful of beans, stand them up on cutting board, holding loosely so they will fall down and have ends ends aligned, then trim. Repeat with other end.) Cut mushrooms into slices 1/2 inch thick.
Put cut beans and mushrooms into a Ziploc bag or plastic bowl. Whisk together olive oil and balsamic vinegar and pour over, then squeeze bag or stir so all the beans and mushrooms are lightly coated with the mixture. Arrange on large cookie sheet, spreading them out well so beans and mushrooms are not crowded. Roast 20-30 minutes, starting to check for doneness after 20 minutes. Cook until beans are tender-crisp, mushrooms are cooked, and all liquid on the pan from mushrooms has evaporated. Season beans to taste with salt and fresh ground pepper, then sprinkle with finely grated Parmesan. Serve hot.
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